Hideshi Yamane

Nonlinear wave equations and singular solutions

We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface, which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed.

Nonlinear Cauchy problems with small analytic data

We study the lifespan of solutions to fully nonlinear Cauchy problems with small real- or complex-analytic data. Our proofs are based on the method of majorants and the fixed point theorem for a contraction mapping.

Ramified Cauchy problem for a class of Fuchsian operators with tangent characteristics

Abstract We consider a ramified Cauchy problem for Fuchsian operators of the form a(x,D)=x 0 (D 0 +qx 0 q−1 D 1 )D 0 + ∑ j=1 n a j (x)D j +b(x) .

Fuchsian Cauchy Problem for Entire Functions

Non-characteristic Cauchy problems have been solved in spaces of entire functions by several authors. See [2], [3] and [4]. On the other hand, [1] proved the unique solvability of a local Fuchsian (characteristic) Cauchy problem.

Characteristic Cauchy problems in the complex domain

Garding-Kotake-Leray showed that in a certain characteristic Cauchy problem

Global Fuchsian Cauchy Problem

Pu = \upsilon \in o\;(the\;sheaf\;of\;holomorphic\;functions)

Nonlinear Singular First Order Partial Differential Equations Whose Characteristic Exponent Takes a Positive Integral Value

with zero Cauchy data on a hypersurface S, u can be ramified. Moreover, u is of the form